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Abstract Algebra

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Minimal Polynomial: Definition and Properties

ABSALG-EL7UJK

Let $F/K$ be a field extension with $[F: K]$ finite and let $K[x]$ be the ring of polynomials in $x$ with coefficients in $K$.

The minimal polynomial of $\alpha\in F$ over $K$ is the monic polynomial $P(x)\in K[x]$ of least degree with $P(\alpha)=0$.

"Monic" means that $P(x)$ has leading coefficient equal $1$.

Let $(P(x))$ be the ideal in $K[x]$ generated by $P(x)$.

Which of the following is NOT a property of the minimal polynomial $P(x)$ of $\alpha\in F$ over $K$?

A

$P(x)$ is unique.

B

$P(x)$ is irreducible.

C

The roots of $P(x)$ are all in $K(\alpha)$.

D

$[K(\alpha):K]=\deg(P)$.

E

$K[x]/(P(x))$ is a field isomorphic to $K(\alpha)$.

F

$K[x]/(P(x))$ is a ring isomorphic to $K[\alpha]$.